Proving Lipschitz Condition for F with $\| \cdot \|_{1}$ Norm

[island-boy]

I need to prove that the function F is Lipschitz, using the
$$\| \cdot \|_{1}$$ norm.

that is, for
$$t \in \mathbb{R}$$
and
$$y, z \in Y(t) \in \mathbb{R}^{2}$$

I must show that
$$\|F(t, y) - F(t, z)\|_{1} < k|y-z|$$

F(t, Y(t)) is given as

$$F(t, Y(t)) = \left( \begin{array}{cc} y' \\ \displaystyle{-\frac{g}{L}\sin(y)}\end{array} \right)$$

my only other given is that
y"(t) = -g/L [sin y(t)]
where g and L are constants.

Now if my calculations are correct, I only need to show that the following is true:

$$\|[\frac{g}{L}(\cos y(t) - \cos z(t)] - [\frac{-g}{L} (\sin y(t) - \sin z(t)] \|_{1} < K|y-z|$$

$$|\frac{g}{L}(\cos y(t) - \cos z(t)| + |-\frac{-g}{L} (\sin y(t) - \sin z(t)| < K|y-z|$$

however, I don't know how to prove the above inequality.
I know that the absolute values of both cos and sin are less than or equal to one, but I don't know if that is helpful.

 

[island-boy]

anyone? help please. thanks!

 

[island-boy]

ok, please ignore this question,I think I may have solved it after I was able to prove |siny| <= |y| in the other thread.

thanks for reading.